How many ways can four points be arranged in a plane so that the six distances between pairs of points take on only two different values?

(In reply to by )

The "point on the near side" would not form an equilateral triangle with the two nearest points.  The whole is more a kite shape than a rhombus.  Consider the fourth point, after the three that make up the equilateral triangle, to be on a circle centered on one of those three points, and on the extended median or altitude from that point.  It's distance from the nearer two points is less than the side of one of the equilateral triangles.  For the rhombus, the different distance would be that to the far point on the triangle.

Posted by Charlie on 2004-11-22 00:02:49